
where \(\mathbf {V}_L=\mathbf {v}_L\mathbf {v}_L^H\) . Note that the rank-one constraint on \(\mathbf {V}_L\)  is dropped in problem (REF ). Fortunately, problem (REF ) is a standard convex semidefinite program, which can be solved by some optimization tools, i.e., CVX [1]}.
Yet, \(\mathrm {rank}(\mathbf {V}_L^{\mathrm {opt}})\)  cannot be always guaranteed, where \(\mathbf {V}_L^{\mathrm {opt}}\)  is the optimal solution to problem (REF ). In this context, it is difficult to obtain the optimal phase-shift vector
\(\mathbf {v}_L^{\mathrm {opt}}\)  from \(\mathbf {V}_L^{\mathrm {opt}}\) . To solve this problem, the following Gaussian randomization approach is applied to \(\mathbf {V}_L^{\mathrm {opt}}\)  [2]}:
